Seifert fibering operators in 3d $$ \mathcal{N}=2 $$ theories

Abstract

We study 3d N = 2 supersymmetric gauge theories on closed oriented Seifert manifolds-circle bundles over an orbifold Riemann surface-, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the "fibering operators." These operators are half-BPS line defects, whose insertion along the S 1 fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space L(p, q) b with rational squashing parameter b 2 ∈ Q, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the threedimensional holomorphic blocks.

Expanding 3d $$ \mathcal{N} $$ = 2 theories around the round sphere

“…From the comparison between eq. (3.19) and the explicit expression for H α and F α given in [25], it is straightforward to check that…”

“…A careful treatment of these terms can be found e.g. in [22,25]. We will not keep track of them but instead will define the partition function Z b only up to an overall factor…”

“…For the case when the cycle [S 1 ] along the fiber is nontrivial in H 1 (M g,p , Z 2 ) (i.e. p is even), there are two types of supersymmetric backgrounds depending on the choice of spinstructure along the S 1 [25]. Depending on the discrete choice, ν R can be either 0 (periodic boundary condition for fermions) or 1/2 (anti-periodic boundary condition).…”

“…One simpler way to understand the second equality is using a factorization of refined twisted index on S 2 as explained in [25]. The refined index is defined as 38) and is known to have the following factorization property [46,47]…”

Universal 3d Cardy block and black hole entropy

“…One can also define an index for a topologically twisted theory on a circle fibered over a Riemann surface of genus g [60][61][62][63][64]. If g = 0, i.e., if the Riemann surface is a sphere, one can refine the index by turning on the angular momentum fugacity.…”